Through the variable substitution and separated variable methods, this study develops a two‐dimensional (2‐D) axisymmetric diffusion analytical solution for organic pollutants in a circular‐shaped cutoff wall (CCW) system for the first time, which can more precisely and reasonably simulate the diffusion behaviors in “circular‐shaped” vertical barriers. Then, the proposed analytical solution's reasonableness is verified by comparing it with an existing analytical solution and a corresponding finite‐difference solution. Meanwhile, the comparison suggests that this solution will degrade to a 2‐D diffusion analytical solution when the pollution source radius is large enough. Furthermore, the presented analytical solution can also be simplified to a one‐dimensional axisymmetric diffusion analytical solution, or to the axisymmetric diffusion analytical solutions in a single‐layered medium. These exact analytical solutions can not only be applied to study axisymmetric diffusion behaviors under specific scenarios, but also be used to validate other complex numerical models. Last, a case study is conducted to investigate the impacts of pollution source concentration distribution, CCW horizontal thickness, and defined equivalent diffusion coefficient on the barrier performance. Overall, the proposed analytical solutions and obtained diffusion laws in this study can provide guidance for the service effect assessment and the engineering design of cutoff walls.

中文翻译：

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圆形截留壁系统中有机污染物轴对称扩散的分析解


通过变量替代和分离变量方法，本研究首次开发了圆形截留墙 （CCW） 系统中有机污染物的二维 （2‐D） 轴对称扩散分析解决方案，可以更精确、更合理地模拟“圆形”垂直屏障中的扩散行为。然后，通过与现有解析解和相应的有限差分解进行比较，验证所提解析解的合理性。同时，比较表明，当污染源半径足够大时，该溶液将降解为 2-D 扩散分析解。此外，所提出的解析解也可以简化为一维轴对称扩散解析解，或简化为单层介质中的轴对称扩散解析解。这些精确的解析解不仅可以应用于研究特定场景下的轴对称扩散行为，还可以用于验证其他复杂的数值模型。最后，进行了案例研究，以调查污染源浓度分布、CCW 水平厚度和定义的等效扩散系数对屏障性能的影响。综上所述，本研究提出的解析解和得到的扩散规律可为防断墙的服务效果评估和工程设计提供指导。